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Chapter 9: Problem 22

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$(\mathbf{a}+\mathbf{b})+\mathbf{c}$$

Short Answer

Expert verified
The resultant vector is \(\langle 13, 6 \rangle\).

Step by step solution

01

Vector Addition of a and b

First, let's find the resultant vector of adding \( \mathbf{a} \) and \( \mathbf{b} \). The addition of two vectors \( \mathbf{a} = \langle 2, 3 \rangle \) and \( \mathbf{b} = \langle 5, 4 \rangle \) is performed by adding their respective components:\[\mathbf{a} + \mathbf{b} = \langle 2 + 5, 3 + 4 \rangle = \langle 7, 7 \rangle\]
02

Add the Resultant to Vector c

Now that we have \( \mathbf{a} + \mathbf{b} = \langle 7, 7 \rangle \), add this vector to \( \mathbf{c} = \langle 6, -1 \rangle \). Again, add the corresponding components of the vectors:\[(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \langle 7, 7 \rangle + \langle 6, -1 \rangle = \langle 7 + 6, 7 + (-1) \rangle = \langle 13, 6 \rangle\]
03

Conclusion

The result of the vector operation \((\mathbf{a} + \mathbf{b}) + \mathbf{c}\) is a new vector \(\langle 13, 6 \rangle\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vectors
Vectors are mathematical objects used to represent quantities that have both magnitude and direction. In everyday life, common examples of vectors include forces, velocities, and displacements. Each vector is defined by two or more components. These components are often written as an ordered pair or triple, representing different dimensions.
For example, in two-dimensional space, a vector
  • can be written as \( \mathbf{a} = \langle x, y \rangle \) where \( x \) and \( y \) are the components
  • represents movement or direction on a plane.
Vectors are fundamental in physics and engineering, as they simplify the representation of physical phenomena. They provide a handy way to perform complex calculations like forces acting at various angles in physics.
Vector Components
The components of a vector are its projections along the coordinate axes. Let's take the vector \( \mathbf{a} = \langle 2, 3 \rangle \) as an example.
Here, the numbers 2 and 3 are its components.
  • The first component (2) describes the horizontal change.
  • The second component (3) describes the vertical change.
The components tell us where the vector is heading and how far it will go in that direction.
When adding vectors, each component of one vector is simply added to the corresponding component of the other vector. Think of it like combining steps in each direction. This way, you get a new path that correctly reflects both original paths combined.
Resultant Vector
A resultant vector is the vector result of adding two or more vectors together. It represents the combined effect of all those vectors.
For example, if a force of 5 N east and a force of 3 N north act on an object simultaneously, the resultant vector can be found using vector addition.
Let's see how this concept applies in our given problem:
  • Start by adding vectors \( \mathbf{a} = \langle 2, 3 \rangle \) and \( \mathbf{b} = \langle 5, 4 \rangle \).
  • Their resultant is \( \langle 7, 7 \rangle \), showing the combination of both forces.
  • Then, add this result to vector \( \mathbf{c} = \langle 6, -1 \rangle \).
  • This gives us the final resultant vector \( \langle 13, 6 \rangle \).
The resultant vector provides a single vector that has the same effect as the individual vectors acting together. It's a crucial concept for simplifying problems involving multiple vector quantities.

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Most popular questions from this chapter

Let \(\theta\) (where \(0 \leq \theta \leq \pi\) ) denote the angle between the two nonzero vectors \(\mathbf{A}\) and \(\mathbf{B}\). Then it can be shown that the cosine of \(\theta\) is given by the formula $$\cos \theta=\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}||\mathbf{B}|}$$ (See Exercise 77 for the derivation of this result.) In Exercises \(65-70,\) sketch each pair of vectors as position vectors, then use this formula to find the cosine of the angle between the given pair of vectors. Also, in each case, use a calculator to compute the angle. Express the angle using degrees and using radians. Round the values to two decimal places. (a) \(\mathbf{A}=\langle 7,12\rangle\) and \(\mathbf{B}=\langle 1,2\rangle\) (b) \(\mathbf{A}=\langle 7,12\rangle\) and \(\mathbf{B}=\langle-1,-2\rangle\)

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. (a) \(x=3 t+2, y=3 t-2,(-2 \leq t \leq 2)\) (b) \(x=3 t+2, y=3 t-2,(-3 \leq t \leq 3)\)

Compute the distance between the given points. (The coordinates are polar coordinates.) $$\left(3, \frac{5 \pi}{6}\right) \text { and }\left(5, \frac{5 \pi}{3}\right)$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=2 \cos t+\cos 2 t, y=2 \sin t-\sin 2 t, \quad 0 \leq t \leq 2 \pi[\) This curve is the deltoid. It was first studied by the Swiss mathematician Leonhard Euler \((1707-1783) .]\)

Find a unit vector having the same direction as the given vector. $$\langle 4,8\rangle$$
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